Consider a geometrically rational variety \(X\) , smooth and
projective over a field \(k\) .
Is \(X\)  rational over \(k\) ? A necessary condition is that \(X(k)\ne \emptyset \) ,
which is sufficient in dimension one, as well as for quadric hypersurfaces
and Brauer-Severi varieties of arbitrary dimension.
When the dimension of \(X\)  is at most two, rationality over \(k\) 
was settled by work of Enriques, Manin, and Iskovksikh [1]}.
Rationality is encoded in the Galois action on the geometric
Néron-Severi group – varieties with rational points that are
`minimal' in the sense of birational geometry need not be
rational. In dimension three,
recent work [2]}, [3]}, [4]}, [5]}, [6]}
has clarified the criteria for rationality: one also needs to take
into account principal homogeneous spaces over the intermediate
Jacobian, reflecting which curve classes are realized over the ground
field.
The case of complete intersections of two quadrics was an
important first step in understanding the overall structure
[7]}; rationality in dimension three
is equivalent to the existence of a line over \(k\)  [2]}, [6]}, [4]}.
